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Featuretheorems – after all, the HOL Light and Isabelle Librariescontain Tom Hales’ proof of the Kepler Conjecture and theCoq library contains the proof of Feit-Thomson’s Odd-OrderTheorem.Note that once a set of views has been found by the viewfinder,we can generate the induced theorems in all accessibletheories and iterate the view finding process, which might findmore views with the induced theorems.Figure 5. Searching for Induced Knowledge Itemsgraph – contains enough information to generate an explanationof the reason + is associative on Z. In essence, thisexperimental search engine searches the space of (induced)mathematical knowledge rather than just the space of mathematicaldocuments.Making Bourbaki AccessibleIdeas like the ones above can solve one of the problems withthe Bourbaki book series, which is written in such a very conciseand modular manner that it can only be understood if onehas all the previous parts in memory. We have extracted thetheory graph underlying the first 30 pages of Algebra I [2].It contains 51 theories, 94 inclusions and 10 views. The theoriescontain 82 Symbols, 38 axioms, 30 theorems and 17definitions [13]. For knowledge items higher up in the graph,there is no (single) place in the book which states all theiraxioms or properties. With the MMT API, we can generateflattened descriptions for reference and with FlatSearch wecan search for their properties. We conjecture that the simpleexplanation feature from Figure 5 can be extended into a“course generator” that generates a self-contained – modulothe reader’s prerequisite knowledge – document that explainsall aspects necessary for understanding the search hits. Withtheory graph technology, Bourbaki’s Elements can be read byneed, as a foundational treatise should be, instead of beingrestricted to beginning-to-end reading.Uncovering TheoremsWe can now come back to the discussion on the OBB from theintroduction. One way to overcome the problem of missinginterpretations is to systematically search for them – after all,machine computation is comparatively cheap and the numberof potential theory morphisms is bounded by the squareof the number of theories times the number of symbols intheories. In particular, the MathWebSearch formula searchengine [10] can efficiently search for substitutions (which areessentially the same as the symbol-to-expression mappingsof theory morphisms). We have explored this idea before thetheory graph technology was fully developed, and the Theo-Scrutor system [17] found a considerable number of simpleviews in the Mizar library. While these were relatively syntacticand obvious – after all, for a view finder to work, theproofs for the proof obligations have to be part of the libraryalready – they had not previously been noticed because of theOBB, even though the Mizar project has a “library committee”tasked with finding such linkages. We expect that oncethe OAF is sufficiently stable, a renewed experiment will yieldmany more, and more interesting views and possibly novelRefining Theory GraphsFinally, even partial views – which should be much more numerousthan total ones in a theory graph – can be utilised. Saywe have the situation below with two theories S and T, a partialtheory morphism S −→ σT with domain D and codomainC, and its partial inverse δ.SDσδTCThen we can pass to the following more modular theorygraph, where S ′ := S \D and T ′ := T\C. In this case wethink of the equivalent theories D and C as the intersection oftheories S and T along σ and δ. Note that any views out of Sand T now have to be studied, if they can be pulled back to Cand D.S ′ T ′δσσDCδWe have observed that many of the lesser known algebraicstructures in Bourbaki naturally arise as theory intersectionsbetween better known structures. We hope to explain the remainingones via other category-theory-inspired theory graphtransformation operations.Note that operations like theory intersections apply theorygraph technology to the problem of theory graph maintenance,which is itself a problem greatly hampered by the OBBwithout MKM techniques.5 ConclusionIn this paper we have explored opportunities to lift the onebrainbarrier in mathematics, which limits the application ofmathematical knowledge both inside the mathematical domainas well as in other disciplines. We propose that the wayforward is to employ computer systems that can systematicallyexplore immense knowledge spaces, if these are representedin sufficiently content-oriented formats. Together withthe creation, curation and application of digital mathematicallibraries (DMLs), this is one of the central concerns of thenew field of Mathematical Knowledge Management (MKM).We have presented the theory graphs approach as a representationparadigm for mathematical knowledge that allowsus to make its modular and highly networked structure explicitand therefore machine-actionable. We have seen thattheory graphs following the “little theories” approach containthe information structures necessary to explain – and thus ultimatelysupport by computer – many mathematical practices26 EMS Newsletter June 2014